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### Tutorial 5

Generating Functions & Sum of Independent Random Variables

Generating Functions

- Generating functions are tools for studying distributions of R.V.’s in a different domain. (c.f. Fourier transform of a signal from time to frequency domain)
- Moment Generating Function gX (t)=E[etX]
- Ordinary Generating Function hX (z)= E[ZX]
- o.g.f. is also called z-transform which is applied to discrete R.V’s only.

z-transform

- We illustrate the use of g.f.’s by z-transform:
- Let a non-negative discrete r.v. X with p.m.f.

{pk, k = 0,1,…}, z is a complex no.

- The z-transform of {pk} is

hX(z) = p0 + p1z + p2z2+ ……

= pkzk

- It can be easily seen that

pkzk = E[zX]

z-transform

- We can obtain many useful properties of r.v. X from hX(z).
- First, we can observe that
- hX(0) = p0 + p10+ p202+ …… = p0
- hX(1) = p0 + p11+ p212+ …… = 1
- By differentiate hX(z), we can get the mean and variance of X.

Mean by z-transform

- Put z = 1, we get
- hX’(1) is the mean of of X.
- Similarly,

Variance by z-transform

- E[X2] is called the 2nd moment of X.
- In general, E[Xk] is called the k-th moment of X. We can get E[Xk] from successive derivatives of hX (z).
- Since Var(X) = E[X2] - E[X]2, we get

Example - Bernoulli Distr.

- Find the mean and variance of a Bernoulli distr. by z-transform.

P(X=1) = p, P(X=0) = 1-p

Example - Bernoulli Distr.

- E[X] = hX’(1) = p

Finding pj from g(t) and h(z)

- If we know g(t), then we know h(z), then we can find the pj :

p.d.f. of sum of R.V.’s

- Let X , Y be 2 independent continuous R.V.’s
- The cumulative distribution function (c.d.f) of X+Y:

p.d.f. of sum of R.V.’s

- By differentiating the above equation, we obtain the p.d.f. of X+Y:
- fX+Y(a) is the convolution of fX and fY .

m.g.f. of sum of R.V.’s

- On the other hand, the moment generating function of p.d.f. fX is
- The m.g.f. of fX+Y is:

m.g.f. of sum of R.V.’s

- We have obtained an important property:
- If S = X+Y, where X & Y are independent.
- In general, if

p.d.f.

m.g.f.

Two-Armed Bandit Problem

- You are in a casino and confronted by two slot machines. Each machine pays off either one dollar or nothing. The probability that the first machine pays off a dollar is x and that the second machine pays off a dollar is y. We assume that x and y are random numbers chosen independently from the interval [0,1] and unknown to you. You are permitted to make a series of ten plays, each time choosing one machine or the other.

Two-Armed Bandit Problem

- How should you choose to maximize the number of times that you win?
- Strategies described in Grinstead and Snell(P.170):
- Play-the-best (calculate the prob. that each machine will pay off at each stage and choose the machine with the higher prob. )
- Play-the-winner (choose the same machine when we win and switch machines when we lose)

Coursework 01

- Modified two-armed bandit problem:

both unknown prob. vary in a linear manner over the twenty plays,

Pr(payoff at kth play for machine i) = ai + kbi

where ai and bi are constants.

- Make a series of 20 plays
- Design a simple strategy to maximize the number of times that you win

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